Exercise 111.43.10. Let $A$ be a Noetherian ring. Let $I$ be an ideal of $A$. Let $M$ be an $A$-module. Let $M[I^\infty ]$ be the set of $I$-power torsion elements defined by

$M[I^\infty ] = \{ x \in M \mid \text{ there exists an }n \geq 1\text{ such that }I^ nx = 0\}$

Set $M' = M/M[I^\infty ]$. Then the $I$-power torsion of $M'$ is zero. Show that

$\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M').$

Warning: this is not trivial. Hints: (1) try to reduce to $M$ finite, (2) show any element of $\mathop{\mathrm{Ext}}\nolimits ^1_ A(I^ n, N)$ maps to zero in $\mathop{\mathrm{Ext}}\nolimits ^1_ A(I^{n + m}, N)$ for some $m > 0$ if $N = M[I^\infty ]$ and $M$ finite, (3) show the same thing as in (2) for $\mathop{\mathrm{Hom}}\nolimits _ A(I^ n, N)$, (3) consider the long exact sequence

$0 \to \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M[I^\infty ]) \to \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M) \to \mathop{\mathrm{Hom}}\nolimits _ A(I^ n, M') \to \mathop{\mathrm{Ext}}\nolimits ^1_ A(I^ n, M[I^\infty ])$

for $M$ finite and compare with the sequence for $I^{n + m}$ to conclude.

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